Heat the oven to 350 degrees F with the oven racks in the upper-middle and lower-middle positions. Line two half sheet pans with parchment paper.Don't use Silpats; the cookies will spread too much.
Combine the peanut butter and both sugars in a large bowl. Use a wooden spoon to stir together, then smush the mixture on the sides of the bowl several times to incorporate the sugar.
Add the egg, baking soda, vanilla, and salt. Stir until really smooth. (If you'd like to make the dough ahead of time, you can cover the bowl with plastic and refrigerate for up to 24 hours.)
Use a 1 1/2-tablespoon-sized disher to scoop the dough out into 16 rounds, evenly spacing 8 dough rounds per sheet. Using a fork in a mash-and-swipe motion, make cross-hatch patterns in each dough round. After being cross-hatched, each cookie should be about 2 1/2 inches in diameter. (If you really want perfectly smooth cookies, you can roll the dough rounds between your hands into balls before forming the cross-hatch pattern. Me? I like 'em rustic.)
Bake until just set, about 10 minutes, rotating the baking sheets between the racks after about 5 minutes. Let the cookies cool on the sheet pans, set on wire racks, for at least 10 minutes before devouring.